«Oracle Semantics Aquinas Hobor A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy ...»

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4. No support for nonlocal exits. As explained in chapter 2, C minor supports various kinds of nonlocal exits; to reason about these we add parameters R and B to the Hoare tuple. We omit them here since these features are directly related neither to concurrency nor to the core of our new deﬁnition of the Hoare tuple.

5. Miscellaneous. Other changes to improve the presentation.

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We start with safeΨ κ, or predicate-level safe, which lifts the safe notion into the modal substructural logic by “hardcoding” the program Ψ and control κ into the predicate. If

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then the control κ is safe with world w in the context of program Ψ.

Just as in the case of the simpler deﬁnitions, we deﬁne the notion of a predicate P guarding a control κ in the context of a program Ψ with P ⊓Ψ κ, or predicate-level guard. Recall from section 7.3.7 that ⊔ the predicate P ⊂ Q is a safe form of logical implication in the modal logic, informally equivalent to “On any world of equal or greater approximation than the current one, P implies Q”. Therefore

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means that for any world w ′ with level less than or equal to the level of w, if w ′ |= P, then Ψ ⊢ safe (w ′, κ).

Our modal Hoare judgment takes a single argument of type Hargs, which is a tuple of predicate (Γ), predicate (P ), statement (c), and predicate (Q). We must uncurry the arguments so that we will be able to use the higher-order recursion operator µHO, which is how we will “tie the knot” when deﬁning the believe and Htuple predicates.

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cation f : {P }{Q} in Γ, the body of the function f, that is, Ψ(f ), has precondition P and postcondition Q.

The ﬁrst argument of believe is a function H from Hargs to predicate— that is, a function that has the same type as the Hoare tuple. The higher-order recursion operator µHO will “tie the knot” and make sure that H will be the Hoare tuple itself. The second argument to believe is the predicate Γ, and the third argument is the program Ψ; the purpose of believe is to relate Γ to Ψ using H.

First, believe quantiﬁes over all functions f, preconditions P, and postconditions Q. Then for any function speciﬁcation f : {P }{Q} that is implied by Γ—that is, if2

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which is ! (Γ ⇒ true ∗ f : {P }{Q}).

This implication is very unusual for us in that we almost never use the “unsafe” form of implication ⇒; moreover looking at it now we suspect that unsafe implication is not needed in the proofs. One of many projects for cleaning up the Coq proofs in the future will be to replace ⇒ with ⊂ in the Coq deﬁnition of believe.

CHAPTER 10. A MODAL HOARE JUDGMENT AND

ORACULAR SOUNDNESS

 We require the ⊲ operator so that believe is contractive in H.

There is a subtlety in this deﬁnition involving the quantiﬁcation over P and Q and the relationship of them to Γ. We are able to quantify over predicates P and Q because we support full impredicative quantiﬁation. Next, recall from section 7.3.10 that due to the way YES inverts, that for a given function f there are multiple P and Q that will be indistinguishable; all that we know is that they are equivalent at strictly greater approximation. This is another reason for applying the ⊲ operator before we pass P and Q to H.

We instantiate the type parameter α with the type Hargs; then µHO binds the variable H for recursive self reference. Now we provide a function of type Hargs → predicate, which we do with a λ as usual, pattern-matching the arguments (Γ, P, c, Q) of the Hoare tuple.

Now we universally quantify over all programs Ψ, and require

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controls κ, if Q guards κ, then P guards c · κ—the diﬀerence is that Ψ is no longer a free variable in the deﬁnition.

Once our Hoare tuple is deﬁned we can deﬁne a notation Γ ⊢ Ψ for believe Htuple Γ Ψ, which is equal to the believe inside the deﬁnition of Htuple by fold-unfold.

Finally, we deﬁne our “user-level” Hoare tuple Γ ⊢ {P } c {Q} as

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using the notation deﬁned in section 7.3.1, where ⊢ P is shorthand for ∀w. w |= P. A Hoare rule is sound only if it is true for all worlds.

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We are now ready at last to prove the axioms of CSL sound with respect to the Concurrent C minor oracular semantics. Later, in section 10.3, we will connect the oracular semantics to the concurrent operational semantics and thereby achieve an end-to-end result.

The Hoare rules divide into three categories. The ﬁrst category, which is by far the most numerous, covers all of the sequential Hoare rules except for call. The second category contains the call rule and the rules for building up the predicate Γ. The third category coveres the concurrent rules.

CHAPTER 10. A MODAL HOARE JUDGMENT AND

ORACULAR SOUNDNESS

 10.2.1 Sequential rules In section 2.5.3 we presented the sequential Hoare rules of Appel and Blazy in ﬁgure 2.6. Appel and Blazy proved those rules sound in Coq with respect to sequential C minor; their proof was a sizable engineering development, and complex enough without worrying about complexities arising from concurrent computation.

Appel was able to adapt those Coq proofs to our new deﬁnitions without altering their essential structure; very little changed, providing strong evidence that our oracular semantics was able to hide the complexities of concurrency from the metatheory proofs about sequential language features, even in the extremely picky context of a machinechecked proof.

10.2.2 Function call The proof of the call rule, on the other hand, did have to change since it interacts with the new Γ ⊢ Ψ predicate. In fact, the semantics of function call had to change so that the resource map was aged during the call; remember from ﬁgure 8.1 that rule sstep-call does just that.

Here we prove a simpliﬁed CSL call rule where we remove the function arguments, but in the Coq development we prove the rule for the full C minor call statement. We will also assume that P and Q are valid in the sense given in section 7.3.8.

Γ ⊢ Ψ is fashionable, meaning that it holds on all worlds of the same level. Therefore, from w |= (Γ ∗ f : {P }{Q}) ⊢ Ψ, we know w ′′′ |= (Γ ∗ f : {P }{Q}) ⊢ Ψ for some w ′′′ of the same level as w and such that there exists an n such that

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Like all the proofs about sequential features, this proof did not mention anything about concurrency. The ability to prove complicated sequential results that are blind to the fact that they are running in a concurrent context is a major strength of our approach.

Combining proofs about functions The major task for the CSL user is to prove Γ ⊢ Ψ—that is, to prove pre- and postconditions for all of the functions in Ψ. For proving indi

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We use the notation Γ ⊢ Ψ : Γ′ as shorthand for believe′ Γ Ψ Γ′.

Γ ⊢ Ψ : Γ′ means that all the functions in Γ′ are proved correct with respect to their function bodies in Ψ. Since a function body may call all functions (not just those proved correct so far), they may use any of the speciﬁcations contained in Γ. This allows Γ′ to be built up one function at a time.

To help the user prove Γ ⊢ Ψ, we provide the rules in ﬁgure 10.4.

The idea is that we will start with Γ′ as emp, with rule func-nil. Then

CHAPTER 10. A MODAL HOARE JUDGMENT AND

ORACULAR SOUNDNESS

 we will add the functions in Γ to Γ′ one at a time, with rule func-cons.

The user will prove each function body using the rules of CSL. When every function in Γ has been added to Γ′, then the two variants of believe are equivalent as expressed in rule func-believe.

Theorem 10.2. The rules func-nil, func-cons, and funcbelieve are sound with respect to the semantic deﬁnitions of Γ ⊢ Ψ and Γ ⊢ Ψ : Γ′.

Proof. Rule func-nil is vacuously true since emp does not contain any functions. Rule func-believe is true immediately from the deﬁnitions. For rule func-cons, believe′ is quantifying over all of the functions in Γ′ plus the new function f. For all of the functions in Γ′, we use the premise Γ ⊢ Ψ : Γ′. For the new function f, we need to prove ⊢ ⊲ Htuple(Γ, P, Ψ(f ), Q), which follows immediately since for any P, (⊢ P ) → (⊢ ⊲ P ).

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The CSL user thus proves the bodies of all his functions with respect to Γ using the rules of CSL, and then uses the rules in ﬁgure 10.4 to prove Γ ⊢ Ψ.

10.2.3 Concurrent rules

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respect to the oracular semantics.

Theorem 10.3. The rules of Concurrent Separation Logic for the concurrent statements of Concurrent C minor as given in ﬁgure 4.1 are sound with respect to the semantics of the Hoare tuple deﬁned in ﬁgure 10.3.

Proof. All of the concurrent statements of Concurrent C minor are handled by the consult partial function. Therefore, our job is to show that the preconditions of the rules guarantee that consult does not get stuck and that afterwards the postconditions hold. Since in the deﬁnition of safe we universally quantify over all oracles, our proofs must hold

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The oracular consult partial function was deﬁned in section

9.3 with three cases. In the ﬁrst case, Ω-invalid, we handle invalid oracles, where there is no concurrent machine state compatible with the oracle. If the oracle is invalid, we loop endlessly. Since looping endlessly is safe regardless of the precondition, and since a postcondition is possible given an inﬁnite loop, we can easily prove the Hoare rules if we have been given an invalid oracle by the universal quantiﬁcation.

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consistency requirements given in section 8.4. This will be one of our major tasks in the case analysis below.

Assuming that this ﬁrst task is done and therefore the preconditions are good enough to guarantee that the concurrent step relation is able to step from S to S ′, the consult function cannot get stuck. Therefore, the remaining task in the soundness proof is to prove the postcondition of the CSL rule. The oracle classically decides whether control will ever return to the thread by branching on the StepOthers relation.

In the ﬁrst case, Ω-diverges, control will never return and the machine enters an inﬁnite loop. As before, this makes it possible to prove any postcondition.

In the ﬁnal case, Ω-steps, control returns after running the other threads. In this case it is necessary to use the precondition of the CSL rules to prove their postcondition by doing induction on the StepOthers relation.

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An examination of the make lock case of the consult function deﬁned in ﬁgure 8.2 will make clear that the consult will not get stuck and will guarantee the postcondtion. The concurrent machine will run cstep-seq to execute this instruction. Since the new lock is created locked, it is simple to show that the machine is still consistent since we only require complex properties to

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An examination of the free lock case of the consult function deﬁned in ﬁgure 8.2 will make clear that the consult will not get stuck and will guarantee the postcondtion. The concurrent machine will run cstep-seq to execute this instruction. Since we have removed a lock from the total resouce map φT, it is easy to show consistency of the new state because we can quantify over one fewer lock.

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the consistency requirements on lock pools it is simple to show a postcondition of ⊲ tightly R, but this is not good enough to prove our CSL rule, since we need tightly R, which is stronger. Accordingly, we need to make a more complex induction, where we argue that since the time when the lock was unlocked, we have already context switched at least once, and therefore we have moved from ⊲ tightly R to tightly R.

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An examination of the preconditions of the CSL rule will demonstrate that the concurrent machine will run cstep-unlock to execute this instruction. Proving the postcondition of emp is not diﬃcult. The diﬃculty is in showing that the new machine state is consistent, and in particular that the rules for the lock pool are

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5. Γ ⊢ {f : {P }{Q}∗validly precisely P } forkf {f : {P }{Q}} An examination of the preconditions of the CSL rule will demonstrate that the concurrent machine will run cstep-fork to execute this instruction. It is not diﬃcult

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oracle semantics to the concurrent semantics in section 10.3, we will have to prove that the has been properly started in the preservation theorem.

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